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PHYSICS A: MAT

HEMATICAL AND

GENERAL

J. Phys. A: Math. Gen. 38 (2005) 55315546

doi:10.1088/0305-4470/38/24/007

Unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation
O N Kirillov, A A Mailybaev and A P Seyranian
Institute of Mechanics, Moscow State Lomonosov University, Michurinskii pr. 1, 119192 Moscow, Russia E-mail: kirillov@imec.msu.ru, mailybaev@imec.msu.ru and seyran@imec.msu.ru

Received 1 November 2004, in final form 8 April 2005 Published 1 June 2005 Online at stacks.iop.org/JPhysA/38/5531 Abstract The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied. PACS numbers: 03.65.Vf, 32.60.+i

1. Introduction Since the papers by Von Neumann and Wigner (1929) and Teller (1937), it is known that the energy surfaces in quantum physics may cross forming two sheets of a double cone: a diabolo. The apex of the cone is called a diabolic point, see Berry and Wilkinson (1984). This kind of crossing is typical for systems described by real symmetric Hamiltonians with at least two parameters and Hermitian Hamiltonians depending on three or more parameters. From a mathematical point of view, the energy surfaces are described by eigenvalues of real symmetric or Hermitian operators dependent on parameters, and the diabolic point is a point of a double eigenvalue with two linearly independent eigenvectors. In modern problems of quantum physics, crystal optics, physical chemistry, acoustics and mechanics, it is important to know how the diabolic point bifurcates under arbitrary complex perturbations forming topological singularities of eigenvalue surfaces like a double coffee filter with two exceptional points or a diabolic circle of exceptional points, see e.g. Mondragon and Hernandez (1993, 1996), Shuvalov and Scott (2000), Keck et al (2003), Berry and Dennis (2003), Korsch and Mossmann (2003) and Berry (2004). In our preceding companion paper (Seyranian et al 2005), a general theory of coupling of eigenvalues for complex matrices of arbitrary dimension smoothly depending on multiple real parameters was presented. Two kinds of important singularities were mathematically
0305-4470/05/245531+16$30.00 © 2005 IOP Publishing Ltd Printed in the UK 5531

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classified: the diabolic points (DPs) and the exceptional points (EPs). DP is a point where the eigenvalues coalesce, while corresponding eigenvectors remain different (linearly independent), and EP is a point where both eigenvalues and eigenvectors coalesce forming a Jordan block. General formulae describing coupling and decoupling of eigenvalues, crossing and avoided crossing of eigenvalue surfaces were derived. Both the DP and EP cases are interesting in applications and were observed in experiments, see Ramachandran and Ramaseshan (1961), Dembowsky et al (2001, 2003) and Stehmann et al (2004). In the present paper following the theory developed in Seyranian et al (2005) we study effects of complex perturbations in multiparameter families of real symmetric and Hermitian matrices. In the case of real symmetric matrices we study the unfolding of eigenvalue surfaces near a diabolic point under real and complex perturbations. The origination of singularities such as a `double coffee filter' and a `diabolic circle' is analytically described. Unfolding of a diabolic point of a Hermitian matrix under an arbitrary complex perturbation is analytically treated. We emphasize that the unfolding of eigenvalue surfaces is described qualitatively as well as quantitatively by using only the information at the diabolic point, including eigenvalues, eigenvectors and derivatives of the system matrix taken at the diabolic point. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied. Asymptotic formulae for the metamorphoses of these surfaces depending on properties of a crystal are established and discussed in detail. Singular axes for general crystals with weak absorption and chirality are found. A new explicit condition distinguishing the absorption-dominated and chirality-dominated crystals is established in terms of components of the inverse dielectric tensor. Numerical examples are given to illustrate the general theory. 2. Asymptotic expressions for eigenvalues near a diabolic point Let us consider the eigenvalue problem Au = u (1) for an m в m Hermitian matrix A, where is an eigenvalue and u is an eigenvector. Such eigenvalue problems arise in non-dissipative physics with and without time reversal symmetry. Real symmetric and complex Hermitian matrices correspond to these two cases, respectively. We assume that the matrix A smoothly depends on a vector of n real parameters p = (p1 ,...,pn ). Let 0 be a double eigenvalue of the matrix A0 = A(p0 ) for some vector p0 . Since A0 is a Hermitian matrix, the eigenvalue 0 is real and possesses two eigenvectors u1 and u2 . Thus, the point of eigenvalue coupling for Hermitian matrices is diabolic. We choose the eigenvectors satisfying the normalization conditions (u1 , u1 ) = (u2 , u2 ) = 1, (u1 , u2 ) = 0, (2) where the standard inner product of complex vectors is given by (u, v) = m 1 ui v i . i= Under perturbation of parameters p = p0 + p, the bifurcation of 0 into two simple eigenvalues + and - occurs. The asymptotic formula for ± under multiparameter perturbation is (Seyranian et al 2005) ± = 0 + f11 + f22 , p ± 2 f11 - f22 , p 4
2

+ f12 , p f21 , p .

(3)

n 1 Components of the vector fij = fij ,...,fij are k fij =

A ui , uj , pk

(4)

Unfolding of eigenvalue surfaces near a diabolic point

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where the derivative is taken at p0 , and inner products of vectors in (3) are given by a, b = n=1 ai bi . Note that we use different notation for the inner product to distinguish i between the linear complex space C m (round brackets) and linear real space R n (angular brackets). In expression (3), the higher-order terms o( p ) and o( p 2 ) are omitted before and under the square root. Since the matrix A is Hermitian, the vectors f11 and f22 are real and the vectors f12 = f 21 are complex conjugate. In the case of real symmetric matrices A = AT , the vectors f11 , f22 and f12 = f21 are real. The asymptotic expression for the eigenvectors corresponding to ± takes the form (Seyranian et al 2005) u± = ± u1 + ± u2 , ± f12 , p ± - 0 - f22 , p = = . ± ± - 0 - f11 , p f21 , p (5)

Expressions (5) provide zero-order terms for the eigenvectors u± under perturbation of the parameter vector. Now, consider an arbitrary complex perturbation of the matrix family A(p) + A(p). Such perturbations appear due to non-conservative effects breaking symmetry of the initial system. We assume that the size of perturbation A(p) is small, where = A(p0 ) is the Frobenius norm of the perturbation at the diabolic point. Behaviour of the eigenvalues ± for small p and small is described by the following asymptotic formula (Seyranian et al 2005): 11 + 22 f11 + f22 , p + ± = 0 + 2 2 ( f11 - f22 , p + 11 - 22 )2 ± (6) + ( f12 , p + 12 )( f21 , p + 21 ). 4 The quantities ij are small complex numbers of order given by the expression ij = ( A(p0 )ui , uj ). (7) A small variation of the matrix family leads to the following correction of the asymptotic expression for the eigenvectors:
u± = ± u1 + ± u2 , ± f12 , p + 12 ± - 0 - f22 , p - 22 = . = ± ± - 0 - f11 , p - 11 f21 , p + 21

(8)

The ratios + + = - - at the point of coincident eigenvalues + = - . Hence, the eigenvectors u+ = u- coincide, and the point of eigenvalue coupling of the perturbed system becomes exceptional (EP). For some specific perturbations A(p), the coupling point may remain diabolic under the conditions

f12 , p + 12 = 0,

f21 , p + 21 = 0,

f11 - f22 , p + 11 - 22 = 0,

(9)

when both ratios in (8) become undetermined. We observe that an asymptotic description of unfolding of a diabolic singularity due to perturbation of the matrix family requires only the value of A(p) taken at the coupling point p0 . Dependence of the perturbation A on the vector of parameters p near the point p0 is not so important, since it influences higher-order terms. 3. Unfolding of a diabolic singularity for real symmetric matrices Let us assume that A(p) is an n-parameter family of real symmetric matrices. Then its eigenvalues are real. Let 0 be a double eigenvalue of the matrix A0 = A(p0 ) with two real

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Figure 1. A diabolic point in a family of real symmetric matrices.

eigenvectors u1 and u2 . Under perturbation of parameters p = p0 + p, the eigenvalue 0 splits into two simple eigenvalues + and - . The asymptotic formula for ± under multiparameter perturbation is given by equations (3) and (4), where the vectors f11 , f22 and f12 = f21 are real. Then, equation (3) takes the form ± - 0 - f11 + f22 , p 2
2

-

f11 - f22 , p 4

2

- f12 , p

2

= 0.

(10)

Equation (10) describes a surface in the space (p1 ,p2 ,...,pn ,), which consists of two sheets + (p) and - (p). These sheets are connected at the points satisfying the equations ± = 0 +
1 2

f11 + f22 , p ,

f11 - f22 , p = 0,

f12 , p = 0,

(11)

where the eigenvalues couple: + = - . Equations (11) define a plane of dimension n - 2. Thus, the double eigenvalue is a phenomenon of codimension 2 in an n-parameter family of real symmetric matrices (Von Neumann and Wigner 1929). For the two-parameter matrix A(p1 ,p2 ), equation (10) defines a double cone with apex at the point (p0 ,0 ) in the space (p1 ,p2 ,), see figure 1. The point (p0 ,0 ) is referred to as a `diabolic point' (Berry and Wilkinson 1984) due to the conical shape of the children's toy `diabolo'. Let us consider a perturbation A(p) + A(p) of the real symmetric family A(p) in the vicinity of the diabolic point p0 , where A(p) is a complex matrix with the small norm = A(p0 ) . Splitting of the double eigenvalue 0 due to a change of the vector of parameters p and a small complex perturbation A is described by equation (6), which acquires the form c = (x + )2 + (y + )2 - 2 . (12) ± = 0 + µ ± c, In equation (12), the quantities 0 ,x and y are real, 0 = 0 +
1 2

f11 + f22 , p ,

x=

1 2

f11 - f22 , p ,

y = f12 , p ,

(13)

while the small coefficients µ, , and are complex: µ = 1 (11 + 22 ), 2 = 1 (11 - 22 ), 2 = 1 (12 + 21 ), 2 = 1 (12 - 21 ). 2 (14)

Separating real and imaginary parts in equation (12), we find Re2 ( - 0 - µ) - Im2 ( - 0 - µ) = Re c, 2Re( - 0 - µ)Im( - 0 - µ) = Im c, (15)

Unfolding of eigenvalue surfaces near a diabolic point

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where Re c = (Im2 - Im2 - Im2 - Re2 ) + (x +Re )2 + (y +Re )2 , Im c = 2((x +Re )Im + (y +Re )Im - Re Im ). (16) (17)

From equations (15), we get the expressions determining the real and imaginary parts of the perturbed eigenvalues, Re ± = 0 +Re µ ± Im ± = Im µ ± (Re c + Re2 c +Im2 c)/2, (18) (19)

(-Re c +

Re2 c +Im2 c)/2.

Strictly speaking, for the same eigenvalue one should take equal or opposite signs before the square roots in (18), (19) for positive or negative Im c, respectively. Equations (18) and (19) define surfaces in the spaces (p1 ,p2 ,...,pn , Re ) and (p1 ,p2 ,...,pn , Im ). Two sheets of the surface (18) are connected (Re + = Re - ) at the points satisfying the conditions Re c 0, Im c = 0, Im c = 0, Re ± = 0 +Re µ, Im ± = Im µ. (20)

while the sheets Im + (p) and Im - (p) are glued at the set of points satisfying Re c 0, (21)

Note that in the neighbourhood of the intersections (20) and (21) the eigenvalue sheets given by formulae (18) and (19) can be described by the following approximate expressions: Re ± = 0 +Re µ ± Im c Im ± = Im µ ± 2 Im c 2 -1 , Re c Re c < 0; (22) Re c > 0.

1 , Re c

The eigenvalue remains double under the perturbation of parameters when c = 0, which yields two equations Re c = 0 and Im c = 0. Two cases are distinguished according to the sign of the quantity D = Im2 +Im2 - Im2 . (23)

If D > 0, then the equations Re c = 0 and Im c = 0, with expressions (16), (17), yield two solutions (xa ,ya ) and (xb ,yb ), where xa,b = Im Re Im ± Im (Im2 +Im2 +Re2 )(Im2 +Im2 - Im2 ) - Re , Im2 +Im2 (24) ya,b = Im Re Im Im (Im2 +Im2 +Re2 )(Im2 +Im2 - Im2 ) - Re . Im2 +Im2 (25) These two solutions determine the points in parameter space, where double eigenvalues appear. When D = 0, the two solutions coincide. For D < 0, the equations Re c = 0 and Im c = 0 have no real solutions. In the latter case, the eigenvalues + and - separate for all p.

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O N Kirillov et al

( b)

Figure 2. Unfolding of a diabolic point due to complex perturbation.

Note that the quantities Im and Im are expressed by means of the anti-Hermitian part T AN = ( A - A )/2 of the matrix A as ( AN (p0 )u1 , u1 ) - ( AN (p0 )u2 , u2 ) , 2i ( AN (p0 )u1 , u2 ) + ( AN (p0 )u2 , u1 ) Im = , 2i Im = while Im depends on the Hermitian part Im = AH = ( A + A )/2as (27)
T

(26)

( AH (p0 )u1 , u2 ) - ( AH (p0 )u2 , u1 ) . 2i

If D > 0, one can say that the influence of the anti-Hermitian part of the perturbation A is stronger than that of the Hermitian part. If the Hermitian part prevails in the perturbation A, we have D < 0. In particular, D = -Im2 < 0 for a purely Hermitian perturbation A. Let us assume that the vector p consists of only two components p1 and p2 , and consider surfaces (18) and (19) for different kinds of perturbation A(p). Consider first the case D < 0. Then, the eigensheets Re + (p) and Re - (p) are separate, see figure 2(a). Indeed, for D -Re2 the inequality Re c 0 holds for all variations of parameters, see equation (16). In the case when -Re2 < D < 0 the equation Re c = 0 with expressions (13) defines an ellipse in the plane of parameters (p1 ,p2 ). Inside the ellipse we have Re c < 0 and outside Re c > 0. The equation Im c = 0 defines a line in parameter plane. The line and the ellipse have no common points for D < 0 since there are no real solutions of the equation c = 0. Hence, for D < 0 conditions (20) are not fulfilled and the real parts of the eigenvalues avoid crossing. As the size of the complex perturbation decreases ( 0), the two sheets come closer and touch each other at the point (p0 ,0 ) for = 0 forming the

Unfolding of eigenvalue surfaces near a diabolic point

5537

diabolic singularity. The sheets Im + (p) and Im - (p) of the eigensurface (19) intersect along the line Im c/2 = (x +Re )Im + (y +Re )Im - Re Im = 0, Im ± = Im µ, (28)

given by conditions (21). Note that, by using (22), one can show that the angle of intersection of the imaginary eigensheets is small of order and tends to zero as 0. In the case D > 0, the line Im c = 0 and the ellipse Re c = 0 have common points pa and pb where the eigenvalues couple. Coordinates of these points found from equations (13) are pa,b = p0 + -
2 2 2 2f12 xa,b - f11 - f22 ya,b 1 2 2 2 1 1 f12 f11 - f22 - f12 f11 - f22

,

1 1 1 2f12 xa,b - f11 - f22 ya,b 1 2 2 2 1 1 f12 f11 - f22 - f12 f11 - f22

,

(29)

where xa,b and ya,b are defined by expressions (24) and (25). Here we have assumed that the vectors f11 - f22 and f12 are linearly independent. Note that the points pa and pb coincide in the degenerate case D = 0. According to conditions (20) the real eigensheets Re + (p) and Re - (p) are glued in the interval [pa , pb ] of the line Im c/2 = (x +Re )Im + (y +Re )Im - Re Im = 0, Re ± = 0 +Re µ. (30) The surface of real eigenvalues (18) is called a `double coffee filter' (Keck et al 2003). The unfolding of a diabolic point into the double coffee filter is shown in figure 2(b). From conditions (21), it follows that the imaginary eigensheets Im + (p) and Im - (p) are connected along the straight line (28) where the interval [pa , pb ] is excluded, see figure 2(b). According to formulae (22) the angle of intersection of the imaginary eigensheets tends to as the points pa and pb are approached, since Re c goes to zero. At far distances from the interval [pa , pb ], this angle becomes small of order . With the decrease of the size of complex perturbation the interval shrinks and the angle of intersection goes to zero. At = 0 the imaginary parts of the eigenvalues coincide: Im + = Im - = 0. Note that in crystal optics and acoustics the interval [pa , pb ] is referred to as a `branch cut', and the points pa , pb are called `singular axes', see Shuvalov and Scott (2000), Berry and Dennis (2003) and Ramachandran and Ramaseshan (1961). According to equation (8) the double eigenvalues at pa and pb possess only one eigenvector and, hence, they are exceptional points (EPs). Now let us consider the case when the perturbation A(p) is real. In this case µ, , , and hence c = (x + )2 + (y + )2 -
2

(31)

are real quantities. According to (12) the eigenvalues ± are complex conjugate if c < 0 and real if c > 0. The eigenvalues couple for c = 0 forming a set consisting of exceptional points with double real eigenvalues. Consider a system depending on a vector of two parameters p = (p1 ,p2 ). Then the equation c = 0 with expressions (13) and (31) defines an ellipse in parameter plane; c < 0 inside the ellipse and c > 0 outside. Real parts of the eigenvalues are given by the equations c c 0: 0: (Re - 0 - µ)2 - (x + )2 - (y + )2 = - 2 , Re = 0 + µ. (32) (33)

Equation (32) defines a hyperboloid in the space (p1 ,p2 , Re ). Real parts of the eigenvalues ± coincide at the disc determined by equation (33), see figure 3. Imaginary parts of the eigenvalues are c 0: Im = 0, (34)

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Figure 3. A real non-symmetric perturbation of a diabolic point.

c

0:

Im2 + (x + )2 + (y + )2 = 2 .

(35)

The imaginary parts are both zero at the points of the plane (34) surrounding the ellipsoid (35) (`a bubble') in the space (p1 ,p2 , Im ), see figure 3. The eigenvalues couple at the points of the elliptic ring ± = 0 + µ, (x + )2 + (y + )2 = 2 , (36)

consisting of exceptional points, see figure 3. For that reason we call it an `exceptional ring', which is a better name compared with a `diabolic circle' suggested by Mondragon and Hernandez (1993, 1996). Note that for the real symmetric perturbation A, we have = 0 according to equations (7) and (14). Then, the radius of the exceptional ring in equation (36) is zero and the perturbation preserves the diabolic point, only shifting it. Finally, it is instructive to consider deformations of the surfaces (32), (33) and (34), (35) as the real perturbation becomes complex. If the imaginary part of the perturbation Im A is such that D < 0, then the parts of the hyperboloid (32) connected by the disc (33) are separated into the two smooth surfaces described by equation (18). On the other hand, the ellipsoid (35) surrounded by the plane (34) is foliated into two sheets crossing each other along the line Im c = 0, see figure 2(a). Recall that the line Im c = 0 does not intersect the ellipse Re c = 0. When D > 0, the disc (33) foliates into two sheets crossing along the interval [pa , pb ], where the points pa and pb are given by expression (29). As the size of the imaginary part of the perturbation Im A increases, the angle of intersection of real eigensheets grows. In this way, the purely imaginary perturbation deforms the hyperboloid (32) into the double coffee filter (18), see figure 2(b). The ellipsoid (35) surrounded by the plane (34) is transformed into two smooth sheets intersecting along the line Im c = 0, where the interval [pa , pb ] is excluded. The angle of intersection grows as the size of the perturbation Im A increases. 4. Unfolding of a diabolic singularity for Hermitian matrices Let us consider a multiparameter Hermitian matrix A(p). Assume that p0 is a diabolic point, where the matrix A0 = A(p0 ) has a double real eigenvalue 0 with two eigenvectors. The splitting of 0 into a pair of simple real eigenvalues + and - is described by expressions (3), (4), where the vectors f11 and f22 are real and the vectors f12 = f 21 are complex conjugate. By using expression (3), we find ± = 0 ± x 2 + y 2 + z2 , (37)

Unfolding of eigenvalue surfaces near a diabolic point

5539

where 0 ,x ,y and z are real quantities depending linearly on the perturbation of parameters p as follows: 0 = 0 + f11 + f22 , p , 2 y = Re f12 , p , x= f11 - f22 , p , 2 z = Im f12 , p . (38)

The eigenvalues coincide if x = y = z = 0. Thus, if the system depends on three parameters and the real vectors f11 - f22 , Re f12 and Im f12 are linearly independent, the eigenvalues + and - split for any nonzero perturbation p. For more than three parameters, the equations x = y = z = 0 with relations (38) provide a plane of diabolic points in parameter space. This plane has dimension n - 3, which agrees with the well-known fact that the diabolic point is a codimension 3 phenomenon for Hermitian systems (Von Neumann and Wigner 1929, Arnold 1972). Now let us consider a general non-Hermitian perturbation of the system A(p) + A(p), assuming that the size of perturbation at the diabolic point = A(p0 ) is small. The two eigenvalues + and - , which become complex due to non-Hermitian perturbation, are given by asymptotic expressions (6), (7). With the use of the new coordinates (38), we write expression (6)as (39) ± = 0 + µ ± c, where c = (x + )2 + (y + )2 + (z - i)2 , and µ, , , are small complex quantities of order given by expressions (14). The eigenvalues couple (+ = - ) if c = 0. This yields two equations Re c = (x +Re )2 + (y +Re )2 + (z +Im )2 - (Im2 +Im2 +Re2 ) = 0, Im c = 2(Im (x +Re ) +Im (y +Re ) - Re (z +Im )) = 0. (41) (42) (40)

Equation (41) defines a sphere in (x ,y ,z) space with the centre at (-Re , -Re , -Im ) and the radius Im2 +Im2 +Re2 , which are small of order . Equation (42) yields a plane passing through the centre of the sphere. The sphere and the plane intersect along a circle. Points of this circle determine values of parameters, for which the eigenvalues ± coincide. Since c = 0 at the coupling point, expression (8) for the eigenvectors takes the form
u± = ± u1 + ± u2 , ± x+ y +iz + + = . = ± -x - y - iz + -

(43)

Thus, all points of the circle are exceptional points, where the two eigenvectors u- and u+ merge in addition to the coupling of the eigenvalues + and - . By using linear expressions (38), the set of exceptional points is found in the original parameter space p. The exceptional circle in (x ,y ,z) space is transformed into an exceptional elliptic ring in three-parameter space p, see figure 4. Let us consider the plane (42), at which the quantity c is real. By formula (39), the real parts of the eigenvalues ± coincide inside the exceptional ring, where c < 0, and the imaginary parts of ± coincide outside the exceptional ring, where c > 0, see the dark and light shaded areas in figure 4. We see that, under a general complex perturbation, a diabolic point of a three-parameter Hermitian system bifurcates into an exceptional ring. This ring has elliptic shape and grows proportionally to the size of perturbation . The real and imaginary parts of the eigenvalues ± coincide, respectively, inside and outside the exceptional ring in the plane of the ring.

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Figure 4. Unfolding of a diabolic point into an exceptional ring in parameter space.
(a) (b)

Figure 5. Surfaces corresponding to coincident real or imaginary parts of eigenvalues.

Note that for the Hermitian perturbation A, one has Im = Im = Re = 0 as follows from equations (7) and (14). Hence, the radius of the exceptional ring is zero, which means that the Hermitian perturbation just shifts the diabolic point without splitting it. Finally, let us study the stratification of parameter space given by the condition |Re(+ - - )| = const. For problems of quantum mechanics, this difference describes the size of a gap between two adjacent energy levels. By using expression (39), we find (+ - - )2 = 4c. Separating real and imaginary parts in this equation and extracting Im(+ - - ), we get Re4 (+ - - ) - 4Re2 (+ - - )Re c - 4Im2 c = 0, (44) where Re c and Im c are given by the first equalities in (41) and (42). Given a fixed value of |Re(+ - - )|, equation (44) with (41), (42) and (38) defines an ellipsoid in three-parameter space enclosing the exceptional ring, see figure 5(a). A similar analysis provides the equation Im4 (+ - - ) +4 Im2 (+ - - )Re c - 4Im2 c = 0 (45) for a surface given by the condition |Im(+ - - )| = const. In three-parameter space equation (45) defines a hyperboloid surrounded by the exceptional ring, see figure 5(b). 5. Unfolding of optical singularities of birefringent crystals Optical properties of a non-magnetic dichroic chiral anisotropic crystal are characterized by the inverse dielectric tensor , which relates the vectors of electric field E and the displacement D as (Landau et al 1984) E = D. (46)

Unfolding of eigenvalue surfaces near a diabolic point

5541

A monochromatic plane wave of frequency that propagates in a direction specified by a real unit vector s = (s1 ,s2 ,s3 ) has the form n(s) T s r-t , (47) D(r,t ) = D(s) exp i c where n(s) is a refractive index, and r is the real vector of spatial coordinates. With the wave (47) and the constitutive relation (46) Maxwell's equations after some elementary manipulations yield (see, e.g., Berry and Dennis (2003)) 1 (48) D(s) - ssT D(s) = 2 D(s). n (s ) Multiplying equation (48) by the vector sT from the left, we find that for plane waves the vector D is always orthogonal to the direction s, i.e., sT D(s) = 0. By using this condition, we write (48) in the form of the eigenvalue problem (I - ssT ) (I - ssT ) u = u,
-2 T

(49)

where = n , u = D and I is the identity matrix. Since I - ss is a singular matrix, one of the eigenvalues is always zero. Let us denote the other two eigenvalues by + and - . These eigenvalues determine refractive indices n, and the corresponding eigenvectors yield polarizations. The inverse dielectric tensor is described by a complex non-Hermitian matrix = transp + dichroic + chiral . The symmetric part of consisting of the real matrix transp and imaginary matrix dichroic constitute the anisotropy tensor, which describes the birefringence of the crystal. For a transparent crystal, the anisotropy tensor is real and is represented only by the matrix transp ; for a crystal with linear dichroism it is complex. Choosing coordinate axes along the principal axes of transp , we have 1 0 0 (50) transp = 0 2 0 . 0 0 3 The matrix
dichroic

= i

d 11 d 12 d 13

d 12 d 22 d 23

d 13 d 23 d 33

(51) chiral gives the antisymmetric part of It is determined by the optical activity
11 12 13

describes linear dichroism (absorption). The matrix describing chirality (optical activity) of the crystal. vector g = (g1 ,g2 ,g3 ) depending linearly on s as 0 -g3 g2 0 -g1 , g = s = chiral = i g3 -g2 g1 0

12 22 23

13 23 33

s1 s2 , s3

(52)

where is a symmetric optical activity tensor; this tensor has an imaginary part for a material with circular dichroism. In the present formulation Berry and Dennis (2003) studied the problem of determining the surfaces of the refractive indices as functions of the components of direction vector s. In their paper the reduction to two dimensions was carried out. Our intention here is to consider the original three-dimensional problem with the inverse dielectric tensor taken in its most general form. Unlike Berry and Dennis (2003) we would like to study the unfolding of the eigenvalue surfaces near the diabolic point in terms of the original problem parameters.

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dichroic

First, consider a transparent non-chiral crystal, when matrix A(p) = (I - ssT )
transp

= 0 and = 0. Then the (53)

(I - ssT )

is real symmetric and depends on a vector of two parameters p = (s1 ,s2 ) (see Berry and Dennis (2003) for other ways of introducing two parameters). The third component of the direction 2 2 vector s is found as s3 = ± 1 - s1 - s2 , where the cases of two different signs should be considered separately. Below we assume that three dielectric constants 1 > 2 > 3 are different. This corresponds to biaxial anisotropic crystals. The nonzero eigenvalues ± of the matrix A(p) are found explicitly in the form (Lewin 1994) trace A 1 2 trace(A2 ) - (trace A)2 . (54) ± 2 2 The eigenvalues ± are the same for opposite directions s and -s. By using (50) and (53) in (54), it is straightforward to show that two eigenvalues + and - couple at ± = s0 = (S1 ,S2 ,S3 ), S2 = 0, 0 = 2 ;
2 S1

S1 = ± (1 - 2 )/(1 - 3 ),

(55)

S3 = ± 1 -

,

which determine four diabolic points (for two signs of S1 and S3 ), also called optic axes (Ramachandran and Ramaseshan 1961). The double eigenvalue 0 = 2 of the matrix A0 = A(p0 ), p0 = (S1 , 0) possesses two eigenvectors 0 S3 u1 = 1 , (56) u2 = 0 , -S1 0 satisfying normalization conditions (2). Using expressions (53) and (56), we evaluate the vectors fij with components (4) for optic axes (55) as f
11

= (0, 0),

f

22

= (2(3 - 1 )S1 , 0),

f

12

=f

21

= (0,(3 - 1 )S1 S3 ).

(57)

By using (55) and (57) in (10), we obtain the local asymptotic expression for the cone singularities in the space (s1 ,s2 ,) as
2 22 ( - 2 - (3 - 1 )S1 (s1 - S1 ))2 = (3 - 1 )2 S1 (s1 - S1 )2 + S3 s2 .

(58)

Equation (58) is valid for each of the four optic axes (55). As an example, consider the case of 1 = 0.5,2 = 0.4,3 = 0.1. Conical surfaces (58) are shown in figure 6 together with the e eigenvalue surfaces (54). The two optic xact e axes presented in figure 6 are s0 = (±1/2, 0, 3/2) with the double igenvalue 0 = 2/5; the eigenvalue surfaces for the opposite directions s0 = (±1/2, 0, - 3/2) are exactly the same. Now let us assume that the crystal possesses absorption and chirality. Then the matrix family (53) takes a complex perturbation A(p) + A(p), where A(p) = (I - ssT )(
dichroic

+

chiral

)(I - ssT ).

(59)

Assume that the absorption and chirality are weak, i.e., = dichroic + chiral is small. Then we can use the asymptotic formulae of sections 2 and 3 to describe unfolding of diabolic singularities of the eigenvalue surfaces. For this purpose, we need to know only the value of the perturbation A at the optic axes of the transparent non-chiral crystal s0 .

Unfolding of eigenvalue surfaces near a diabolic point

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Figure 6. Diabolic singularities near optic axes and their local approximations.

Substituting matrix (59) evaluated at optic axes (55) into expression (7), we obtain
d 11 = i22 , d2 d d2 22 = i11 S3 - 2i13 S1 S3 +i33 S1 , d d 12 = -i 23 + 11 S1 + 13 S3 S1 +i 12 - 13 S1 - 33 S3 S3 , d d 21 = -i 23 - 11 S1 - 13 S3 S1 +i 12 + 13 S1 + 33 S3 S3 .

(60)

By using formulae (14), we get
d d2 d d2 µ = i 22 + 11 S3 - 213 S1 S3 + 33 S1 d d2 d d2 = i 22 - 11 S3 +213 S1 S3 - 33 S1 d d = i 12 S3 - 23 S1 , 2 2 = -i 11 S1 +213 S1 S3 + 33 S3 .

2, 2, (61)

We see that µ, and are purely imaginary numbers depending only on dichroic properties of the crystal (absorption). The quantity depends only on chiral properties of the crystal; is purely imaginary if the optical activity tensor is real. Singularities for crystals with weak dichroism and chirality were studied recently in Berry and Dennis (2003). It was shown that the double coffee filter singularity arises in absorptiondominated crystals, and the sheets of real parts of eigenvalues are separated in chiralitydominated crystals. According to the results of section 3, these two cases are explicitly determined by the conditions D > 0 and D < 0, respectively, where D = Im2 +Im2 -Im2 . These conditions are new and important because they provide quantitative definitions of absorption-dominated and chirality-dominated regimes for unfolding of the diabolic singularity in terms of components of the inverse dielectric tensor. Indeed, according to (61), and depend linearly on all the components of the tensor dichroic , while depends linearly on the components ij ,i,j = 1, 3, of the optical activity tensor . Note that according to the sign of the quantity D taken at different optic axes, we can classify crystals by their optic properties. For example, the important case is a chiralitydominated crystal with D < 0 for all four optic axes. Then real parts of the eigenvalues separate for all directions s. There are four optic axes (55), which determine two pairs of opposite space direction ±s0 . It is easy to see that the unfolding conditions coincide for the optic axes given by opposite directions, while these conditions are different for different pairs of optic axes. In

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(b)

Figure 7. Unfolding of singularities near optic axes.

the absorption-dominated case, when diabolic singularities unfold into coffee filters near two opposite optic axes ±s0 = ±(S1 , 0,S3 ), the four exceptional points of eigenvalue coupling ±sa and ±sb (also called singular axes) appear. By using (57)in(13), we obtain the asymptotic formulae s
a,b 1

= S1 +

xa,b , (1 - 3 )S1

s

a,b 2

=

ya,b , (3 - 1 )S1 S3

s

a,b 3

=

1- s

a,b 2 1

-s

a,b 2 2

, (62)

for the components of the vectors sa,b , where xa,b and ya,b are found by using (25) and (61). In particular, for non-chiral crystals, we have = 0. Then d d d (25) yield xa,b = ±Im = ± 12 S3 - 23 S1 and ya,b = Im = ± 22 - d2 33 S1 2. The equation Im c = 0 determines a line of singularities in the p = (s1 ,s2 ). By using (57), (61) in (13), (17), we find this line in the form

expressions (24), expressions (24), d2 d 11 S3 +213 S1 S3 - parameter space

(s1 - S1 )S1 (1 - 3 )Im - s2 S1 S3 (1 - 3 )Im - Re Im = 0.

(63)

a, a, In the absorption-dominated case, line (63) contains two exceptional points pa,b = s1 b ,s2 b corresponding to the singular axes sa,b . A segment between the points pa and pb corresponds to the coincidence of real parts of the eigenvalues Re + = Re - , while imaginary parts of the eigenvalues Im + = Im - merge at points of line (63) outside this segment, see figure 2(b). In the chirality-dominated case, when singular axes do not appear, imaginary parts of the eigenvalues Im + = Im - coincide at points of the whole line (63), see figure 2(a). If the optical activity tensor is real or purely imaginary, then the line of singularities (63) passes through the diabolic point p0 , and the position of this line does not depend on .

Unfolding of eigenvalue surfaces near a diabolic point

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As a numerical example, let us consider a crystal possessing described by tensors (51), (52) with 320 1 i 2 3 1 , = dichroic = 200 200 013

weak absorption and chirality 3 1 2 1 3 1 2 1 . 3

(64)

A corresponding transparent non-chiral crystal is characterized by 1 = 0.5,2 = 0.4,3 = 0.1, and its eigenvalue surfaces with two optic es are presented in figure 6. By using (64)in ax (61), we find that the condition D = 1607000 (4 3 - 5) > 0 is satisfied for the left optic axis s0 = (-1/2, 0, 3/2). Hence, the diabolic singularity bifurcates into a double coffee filter with two exceptional points whose coordinates according to expressions (62)are 1 1 pa = - - -35 + 28 3, 0 , 2 80 11 pb = - + -35 + 28 3, 0 . 2 80 (65)

Local approximations of the eigenvalue surfaces are given by expressions (18), (19), where 35 - 28 3 1 32 6+ 3 2 + (s1 +1/2) + s, s2 . Re c = Im c = - (66) 160 000 25 100 2 2000 Figure 7(a) shows these local approximations compared with the exact eigenvalue surfaces given by (54). For the right optic axis s0 = (1/2, 0, 3/2), the condition D = - 1607000 (4 3+ 5) < 0 is satisfied. Hence, the eigenvalue sheets (for real parts) separate under the bifurcation of the right diabolic singularity. Approximate and exact eigenvalue surfaces are shown in figure 7(b). The approximations are given by expressions (18), (19), where 32 6- 3 35 + 28 3 1 2 + (s1 - 1/2) + s, s2 . Im c = - (67) Re c = 160 000 25 100 2 2000 We observe that the unfolding types are different for different optic axes. As is seen from figure 7, the asymptotic formulae provide an accurate description for unfolding of eigenvalue surfaces near diabolic points. 6. Conclusion Non-Hermitian Hamiltonians and matrices usually appear in physics when dissipative and other non-conservative effects are taken into account. The known examples are complex refractive indices in optics and complex potentials describing the scattering of electrons or x-rays. Traditionally, non-Hermitian matrices appear in physics as a perturbation of Hermitian matrices. As stated in Berry (2004), Hermitian physics differs radically from non-Hermitian physics in the case of coalescence (coupling) of eigenvalues. In the present paper we have studied this important case carefully. We gave an analytical description for unfolding of eigenvalue surfaces due to an arbitrary complex perturbation with the singularities known in the literature as a `double coffee filter' and a `diabolic circle'. As an application of the presented theory, we found explicit asymptotic expressions for the surfaces of refractive indices in crystal optics in a rather general formulation and obtained a simple condition for the origination of a branch cut in the absorption-dominated crystals in terms of the components of anisotropy and optical activity tensors. We emphasize that the developed theory requires only eigenvectors and derivatives of the matrices taken at the singular point, while the size of the matrix and its dependence on parameters are arbitrary. This makes the presented theory powerful and practical for a wide class of physical problems. The given physical example from crystal optics demonstrates the applicability and accuracy of the theory.

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Acknowledgments The work is supported by the research grants RFBR 03-01-00161, CRDF-BRHE Y1-MP-0619 and CRDF-BRHE Y1-M-06-03. References
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